The simplest, lowest-tech option—beyond operating a recorder while you blather over the speakerphone—is a cable: the Olympus TP-8 Telephone Pick-up Microphone for under $20. It doesn't digitally capture from your iPhone. Instead, it has a microphone built into the earpiece. Plug the 3.5mm plug on the other end into a recorder. Hold the iPhone up to your ear to talk normally. The TP-8 captures each side of the conversation from what comes out of the iPhone's ear speaker, while you can still hear the conversation.
Sometimes, when the phone is not picked, the caller is automatically redirected to the voice mail. That means that when someone calls you, he or she will hear some dial tones before going to the voice mail where they could record their message. In Verizon, you can actually set the number of rings before voicemail.

You can also head to Settings → Wi-Fi → and toggle it off. Now try using voicemail.
Hello! Thanks for calling [LinkedPhone’s Customer Success Team]. We are currently helping other customers achieve their goals and want to do the same for you! Please leave a message with your name, number, and a brief description of how we can serve you. We promise to return your call [before close of business today]. In the interim, you can also check out our website at [linkedphone.com] for helpful answers to our most common requests. Thank you.